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Mirrors > Home > ILE Home > Th. List > elabg | GIF version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |
Ref | Expression |
---|---|
elabg.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
elabg | ⊢ (A ∈ 𝑉 → (A ∈ {x ∣ φ} ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2175 | . 2 ⊢ ℲxA | |
2 | nfv 1418 | . 2 ⊢ Ⅎxψ | |
3 | elabg.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
4 | 1, 2, 3 | elabgf 2679 | 1 ⊢ (A ∈ 𝑉 → (A ∈ {x ∣ φ} ↔ ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 |
This theorem is referenced by: elab2g 2683 intmin3 3633 finds 4266 elxpi 4304 ovelrn 5591 indpi 6326 peano5nni 7698 peano5setOLD 9400 |
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